Poincaré inequalities with the Radon measure for differential forms. (English) Zbl 1189.35011

Xing, Yuming; Ding, Shusen

Comput. Math. Appl. 59, No. 6, 1944-1952 (2010).

Summary: We establish the local and global Poincaré inequalities with the Radon measure for the solutions to the nonlinear elliptic partial differential equation for differential forms.

MSC:

35A23	Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35R06	PDEs with measure
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Poincaré inequality; the \(A\)-harmonic equation; differential forms

References:

[1]	Staples, S. G., \(L^p\)-averaging domains and the Poincaré inequality, Ann. Acad. Sci. Fenn, Ser. AI. Math., 14, 103-127 (1989) · Zbl 0706.26010
[2]	Acosta, G.; Durán, R., An optimal Poincaré inequality in \(L^1\) for convex domains, Proc. Amer. Math. Soc., 132, 195-202 (2004) · Zbl 1057.26010
[3]	Agarwal, R. P.; Ding, S.; Nolder, C. A., Inequalities for Differential Forms (September,2009), Springer
[4]	Beckner, W., A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105, 397-400 (1989) · Zbl 0677.42020
[5]	Belloni, M.; Kawohl, B., A symmetry problem related to Wirtinger’s and Poincaré inequality, J. Differential Equations, 156, 211-218 (1999) · Zbl 0954.26005
[6]	Benguria, R. D.; Depassier, C., A reversed Poincaré inequality for monotone functions, J. Inequal. Appl., 5, 91-96 (2000) · Zbl 0949.26006
[7]	Chavel; Feldman, E. A., An optimal Poincaré inequality for convex domains of non-negative curvature, Arch. Ration. Mech. Anal., 65, 263-273 (1977) · Zbl 0362.35059
[8]	Ding, S.; Nolder, C. A., Weighted Poincaré-type inequalities for solutions to the \(A\)-harmonic equation, Illinois J. Math., 2, 199-205 (2002) · Zbl 1071.35520
[9]	Hebey, E., Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds, Trans. Amer. Math. Soc., 354, 1193-1213 (2002) · Zbl 0992.58017
[10]	Wang, Y.; Wu, C., Sobolev imbedding theorems and Poincaré inequalities for Green’s operator on solutions of the nonhomogeneous \(A\)-harmonic equation, Comput. Math. Appl., 47, 1545-1554 (2004) · Zbl 1155.31303
[11]	Xing, Y., Weighted Poincaré-type estimates for conjugate A-harmonic tensors, J. Inequal. Appl., 1, 1-6 (2005) · Zbl 1087.31009
[12]	Ding, S., Two-weight Caccioppoli inequalities for solutions of nonhomogeneous \(A\)-harmonic equations on Riemannian manifolds, Proc. Amer. Math. Soc., 132, 2367-2375 (2004) · Zbl 1127.35021
[13]	Liu, B., \(A_r^\lambda(\Omega)\)-weighted imbedding inequalities for A-harmonic tensors, J. Math. Anal. Appl., 273, 667-676 (2002) · Zbl 1035.46024
[14]	Nolder, C. A., Hardy-Littlewood theorems for \(A\)-harmonic tensors, Illinois J. Math., 43, 613-631 (1999) · Zbl 0957.35046
[15]	Xing, Y., Weighted integral inequalities for solutions of the \(A\)-harmonic equation, J. Math. Anal. Appl., 279, 350-363 (2003) · Zbl 1021.31004
[16]	Cartan, H., Differential Forms (1970), Houghton Mifflin Co.: Houghton Mifflin Co. Boston · Zbl 0213.37001
[17]	Iwaniec, T.; Lutoborski, A., Integral estimates for null Lagrangians, Arch. Ration. Mech. Anal., 125, 25-79 (1993) · Zbl 0793.58002
[18]	Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princton University Press: Princton University Press Princton · Zbl 0207.13501